Image Reconstruction from Projections

Computed Tomography (CT) is a widely used imaging modality that combines X-ray measurements from different angles to produce cross-sectional images of a patient’s body. It plays a crucial role in medical diagnostics, allowing doctors to visualize internal organs, bones, and tissues in great detail. CT reconstruction involves the application of advanced mathematical techniques to convert raw projection data (obtained through X-ray measurements) into images that can be interpreted by medical professionals.

In a typical CT scan, an X-ray source rotates around the patient, projecting X-rays at various angles. Detectors measure the intensity of the X-rays after they pass through the patient’s body. Since different tissues (e.g., bones, muscles, and organs) absorb X-rays to varying degrees, the measurements recorded by the detectors provide information about the internal structure of the body. The goal is to reconstruct a series of 2D images (slices) from these measurements, which together can form a detailed 3D representation of the scanned region.

Mathematically, the problem of CT image reconstruction can be described as an inverse problem, where the challenge is to infer the internal structure (the image) from the observed data (the X-ray projections).

Mathematical Foundations of CT

1. The Radon Transform

The central mathematical tool in CT reconstruction is the Radon transform, which represents the projection of an object’s internal structure along a particular angle. If we define the object as a 2D function $f(x, y)$ representing the X-ray attenuation at each point, the Radon transform gives the line integral of this function along lines passing through the object at various angles.

For an image $f(x, y)$, the Radon transform $R_f(s, \theta)$ at an angle $\theta$ and distance $s$ from the origin is given by:

The Radon transform essentially provides the “projection” of the image $f(x, y)$ along lines perpendicular to the vector $(\cos \theta, \sin \theta)$.

In CT, these projections are measured at multiple angles around the patient. The goal is to recover the original image $f(x, y)$ from the set of projections $R_f(s, \theta)$ taken over different angles $\theta$.

2. Fourier Slice Theorem

One of the most fundamental mathematical principles used in CT reconstruction is the Fourier Slice Theorem (also known as the Central Slice Theorem). This theorem provides a link between the projections of an object and its representation in the frequency domain.

The Fourier Slice Theorem states that the 1D Fourier transform of a projection $R_f(s, \theta)$ is equivalent to a slice of the 2D Fourier transform of the image $f(x, y)$ along a line at the same angle $\theta$. In other words, the Fourier transform of the Radon transform of the image gives the values of the 2D Fourier transform of the image along radial lines in the frequency domain.

Mathematically:

where $\mathcal{F}_1$ is the 1D Fourier transform of the projection, and $\mathcal{F}_2$ is the 2D Fourier transform of the image. The coordinates $(\omega_x, \omega_y)$ are frequency-domain variables related to the angle $\theta$.

Using the Fourier Slice Theorem, it becomes possible to build the 2D Fourier transform of the image by combining the 1D Fourier transforms of the projections obtained at different angles. Once the 2D Fourier transform is assembled, the inverse Fourier transform can be applied to recover the original image.

3. Filtered Back Projection (FBP)

The most commonly used reconstruction algorithm in CT is Filtered Back Projection (FBP). FBP is a two-step process that uses both back projection and filtering techniques to reconstruct an image from the projection data.

• Back Projection: The first step of FBP is back projection, which essentially smears the projection data across the image space. For each projection, the values along the corresponding X-ray paths are spread evenly across the image. However, back projection alone tends to blur the reconstructed image, especially near the center.

  Mathematically, the back-projected image $f_{\text{bp}}(x, y)$ is given by summing the projections at all angles:

  $$f_{\text{bp}}(x, y) = \int_0^\pi R_f(s, \theta) \, d\theta$$

  where $s = x \cos \theta + y \sin \theta$.

• Filtering: To correct the blurring introduced by back projection, the projections are first filtered in the frequency domain using a high-pass filter. This step amplifies the higher-frequency components of the image, which are necessary for preserving sharp edges and fine details.

  The filtered back projection formula is:

  $$f(x, y) = \int_0^\pi \left( R_f(s, \theta) * h(s) \right) \, d\theta$$

  where $h(s)$ is the filter function, typically a ramp filter, applied in the frequency domain.

FBP is computationally efficient and widely used in clinical CT imaging, although it has limitations when the data is noisy or incomplete.

4. Iterative Reconstruction Techniques

While FBP is fast and effective for well-sampled, high-quality data, iterative reconstruction techniques are more flexible and robust in handling noisy or incomplete data. These methods iteratively refine the reconstructed image by comparing the measured projections to those generated from the current image estimate and adjusting the image accordingly.

Two popular iterative reconstruction techniques are:

• Algebraic Reconstruction Technique (ART): ART formulates the reconstruction problem as a system of linear equations, where each equation corresponds to a line integral (a projection). The image is updated iteratively by solving these equations using methods like least-squares optimization.

  The update rule in ART is:

  $$f_{n+1} = f_n + \lambda \frac{g_i – P_i f_n}{\| P_i \|}$$

  where $f_n$ is the current estimate of the image, $g_i$ is the measured projection data, $P_i$ is the projection operator, and $\lambda$ is a relaxation parameter controlling the step size.

• Simultaneous Iterative Reconstruction Technique (SIRT): SIRT simultaneously updates the image using information from all projections in each iteration. This method converges more slowly than ART but is more stable and handles noisy data better.

  The SIRT update rule is:

  $$f_{n+1} = f_n + \lambda P^T (g – P f_n)$$

  where $P^T$ is the transpose of the projection operator $P$.

5. Regularization and Sparse Reconstruction

CT image reconstruction can suffer from noise and artifacts, particularly in low-dose scans or when there are few projections. Regularization techniques are used to improve the quality of the reconstruction by introducing prior knowledge or constraints into the reconstruction process.

• Tikhonov Regularization: Adds a smoothness constraint to the reconstruction to reduce noise. The regularized reconstruction problem is:

  $$f_{\text{reg}} = \arg \min_f \left( \| P f – g \|_2^2 + \alpha \| f \|_2^2 \right)$$

  where $\alpha$ is a regularization parameter that controls the trade-off between data fidelity and smoothness.

• Total Variation (TV) Regularization: Promotes piecewise smoothness, preserving edges while suppressing noise. TV regularization is particularly useful in sparse reconstruction scenarios, where the number of projections is limited.

  The TV regularization problem is:

  $$f_{\text{TV}} = \arg \min_f \left( \| P f – g \|_2^2 + \beta \int |\nabla f| \, dx \right)$$

  where $\beta$ controls the strength of the regularization.

6. Sparse-View CT and Compressed Sensing

With advances in compressed sensing, it has become possible to reconstruct high-quality images from fewer projections, significantly reducing the radiation dose. Compressed sensing exploits the sparsity of images in certain transform domains (such as wavelets) to reconstruct images from undersampled data.

The mathematical formulation of compressed sensing-based reconstruction is:

where $\Psi$ is a sparsifying transform (e.g., wavelet transform) and $\| \cdot \|_1$ denotes the $\ell_1$-norm, which promotes sparsity.

Example: Medical CT Reconstruction

In medical applications, CT scanners rotate around the patient, capturing X-ray projections at many different angles. These projections are then used to reconstruct cross-sectional images, allowing doctors to visualize internal structures in great detail. For example, in a chest CT scan, the system captures multiple projections of the lungs and thorax, and the FBP algorithm reconstructs the 2D image slices that can reveal diseases like lung cancer or pneumonia.